Cauchy distribution

In mathematics, the Cauchy-Lorentz distribution (after Augustin-Louis Cauchy and Hendrik Lorentz) is a continuous probability distribution with two parameters: a location parameter and a scale parameter.^[1]^[2] As a probability distribution, it is usually called a Cauchy distribution. Physicists know it as a Lorentz distribution.

When the location parameter is 0 and the scale parameter is 1, the probability density function of the Cauchy distribution reduces to [math]\displaystyle{ f(x)= 1/[\pi (x^2+1)] }[/math]. This is called the standard Cauchy distribution.^[2]

The Cauchy distribution is used in spectroscopy to describe the spectral lines found there, and to describe resonance.^[3] It is also often used in statistics as the canonical example of a "pathological" distribution, since both its mean and its variance are undefined. The look of a Cauchy distribution is similar to that of a normal distribution, though with longer "tails".^[4]

Estimating the mean (middle column) and standard deviation (right column) through samples from a Cauchy distribution (bottom) and normal distribution (top). The top graphs converge, wheras the bottom graphs jump up and down

Due to this, estimating the mean value may not converge to any single value with more data (law of large numbers) unlike a normal distribution; due to a higher chance of getting extreme values (the tails of a frequency plot).

Cauchy Distribution Media

Probability density function for the Cauchy distribution
Cumulative distribution function for the Cauchy distribution
Observed histogram and best fitting Cauchy density function.
Observed histogram and best fitting normal density function.
A typical trajectory of sample means looks like long periods of slow convergence to zero, punctuated by large jumps away from zero, but never getting too far away. A typical trajectory of sample variances looks similar, but the jumps accumulate faster than the decay, diverging to infinity.
Fitted cumulative Cauchy distribution to maximum one-day rainfalls using CumFreq, see also distribution fitting